Are there uses of linear logic in linguistics?

I've kind of heard of some applications but it's hard for me to see the big picture.

  • 2
    This is the course material from one of the courses I attended. I think your question is too broad to give a more concrete answer.
    – prash
    Commented May 5, 2015 at 19:12

1 Answer 1


Yes, linear logic has close connections with Lambek/categorial grammar. The big picture is basically that, with respect to a Lambek/categorial grammar, a proof of the syntactic category of a phrase of a language corresponds to a proof in the logical (e.g. deductive) sense, and that the grammar itself corresponds to a certain substructural logic. As Morrill (2011) puts it, "Technically, the Lambek calculus L is the multiplicative fragment of a non-commutative intuitionistic linear logic without empty antecedents."

A couple references:

  • Glynn V. Morrill. 2011. Categorial Grammar: Logical Syntax, Semantics, and Processing.

  • Richard Moot and Christian Retoré. 2012. The Logic of Categorial Grammars: A Deductive Account of Natural Language Syntax and Semantics.

Those deal mainly with syntax and talk a lot about linear logic. A good book dealing with semantics is Carpenter 1998, although his discussion of substructural logic (Ch. 5, sec. 4, pp. 169-170) is very limited.

  • Bob Carpenter. 1998. Type-Logical Semantics.

These references are full-length books. The original work developing linear logic, Lambek/categorial grammar, and their connections was mainly done in article form. The classic Lambek article is

  • Joachim Lambek. 1958. The Mathematics of Sentence Structure. The American Mathematical Monthly, 65(3), 154-170.

The connection between the Lambek calculus and natural deduction/substructural logic was, I think, first studied by Johan van Benthem. Consult the books above for specific and further references.

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